nLab Liouville cocycle

Idea

The Liouville cocycle is a degree 1 cocycle in the groupoid cohomology of the action groupoid of the action of conformal rescalings on the space of Riemannian metrics on a surface.

Definition

Recall that given a space $X$ with an action of a group $G$ on it, a $U(1)$-cocycle on the action groupoid $X// G$, i.e. a functor $X//G \to \mathbf{B} U(1)$, can be explicitly described as a function $\lambda: G\times X \to U(1)$ such that

$$\lambda(h g,x)=\lambda(h, g x)\lambda(g,x).$$

Now fix a Riemann surface $\Sigma$ and take as $X$ the space of Riemannian metrics on $\Sigma$, and as $G$ the additive group of real-valued smooth functions on $\Sigma$, acting on metrics by conformal rescaling:

$$(f,g_{i j})\mapsto e^f g_{i j} \,.$$

The Liouville cocycle with central charge $c\in \mathbb{R}$ is the function

$$\lambda: C^\infty(\Sigma,\mathbb{R})\times Met(\Sigma) \to U(1)$$

defined by

$$\lambda(f,g)=exp(\frac{i c}{2} \int_\Sigma(d f\wedge *_g d f +4 f R_g d \mu_g)),$$

where $*_g$ is the Hodge star operator defined by the Riemannian metric $g$, $R_g$ is the scalar curvarure? and $d \mu_g$ is the volume form.

In conformal field theory

In conformal field theory, the Liouville cocycle appears when one moves from genuine representations of 2-dimensional conformal cobordisms to projective representations. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge $c$; when $c\neq 0$, one says that the conformal field theory has a conformal anomaly.